Two years ago I came across a very beautiful geometric construction (Figure 1), on a tee-shirt I received from the Upstate New York math team. Written on the shirt was a surprising result of iterating a certain construction multiple times. The coaches told me that the proof for the result which they had was very long and very complex. Such a beautiful problem deserves a beautiful answer. So, I came up with a very nice proof which took less than a minute to present to the team the following year. Since then I have delved further into the construction, arriving at some very interesting results. This paper proves some of those results which arise from iterating the geometric construction.
"Iterated Perpendicular Constructions from Interior Points on N-gons,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 3
, Article 5.
Available at: http://scholar.rose-hulman.edu/rhumj/vol3/iss1/5